370 A. C. LAND THE ANGLES OF CRYSTALS. 



This we may write, letting x'^ + y^ = r\ aud - - = p, since m" + n' = 1 as 

 usual — 



By combining (5) and (6) we get — 



^1-^^ 



P 



' [x- pyj 



Since in the figure (6) P^ and P^ are symmetrical with regard to the axis 

 of abscissas, we change n into — n, (i. e., p into — ;)) in order to get a^ and 





(^ + i^!/) 



The angle /'? between the circles at S is the angle between the radii of 

 those circles (from S to a^, h^ and a^, b.^ Avith slopes A^ and A.,). If, then, 



tan /5 = - — 



UA^._ (7) 



while 



A =2'-^:^, A, = 2/^:1^; (8) 



' ' X — «., 



then, by substituting for a^, b^, a.^, b.^ their values found above (6), we have — 



^_a = 2 o:^ - 2 pxy - r -\- 1 ^ ^ _ 2 y^ - 2 .ry/p - r^ + 1 . ^ 

 ' 2(rc— pi/j ' ' —2lp(x — py^ 



2(.;+i>2/) ' ^ '■' 2rp(x+py) • J 



Now, eliminating ?^^ and /^^ from (7) and (8), we have — 

 (x—ci^) (x—a^) + (y—b,) (y—b,) = k {y—b,) (x— a.,) (ij—b.,) {x—a^). (10) 



